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Re: Mass And Gravitational Force

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Posted by pmb on February 8, 2000 16:28:12 UTC

: Then along came Einstein. He proposed in the special theory of relativity,

: 1. The speed of light is constant in a given inertial frame of reference. (already proven) : 2. There exist global spacetime frames with respect to which unaccelerated objects move in straight lines at constant velocity

The two postulates of Special Relativity (SR) are

1. The speed of light is the same in all inertial frames of reference. 2. The laws of physics are the same in all inertial frames of reference.

(Although Post. #1 may be derivable from Post. #2. In fact Einstein claimed this was the case in his E=mc2 paper which appeared in the same issue of the journal where the SR paper appeared.

: Special Relativity only works for non-accelerating frames and does not explain gravity. What Einstein wanted was a theory to explain gravity and include accelerating systems, or frames. This led him to the General Theory of Relativity which says that,

: 1. Gravity and Acceleration are the same,

Actually he stated them as being equivalent rather than the same. But the equivalence was for uniformly accelerating frames and for uniform g-fields. This equivalence didn't extend to all gravitational fields, or at least this is what Einstein claimed. One can always tell if they are in a gravitational field if the field isn't uniform.

As Einstein explained Of course we cannot replace any arbitrary gravitational field by a state of motion of the system without a gravitational field, any more than, by a transformation of relativity, we can transform all points of a medium in any kind of motion to rest

Now we might easily suppose that the existence of a gravitational field is only an apparent one. We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes [33].

Space-time regions of finite extent are, in general, not Galilean, so that a gravitational field cannot be done away with by any choice of co-ordinates in a finite region.

: This is very, very important.

: The reasoning goes like this. If you are in a gravitational field you are accelereting.

This isn't how Einstein explained it (exactly). He described it rather differently In a homogeneous gravitational field (acceleration of gravity g) let there be a stationary systems of co-ordinates K, oriented so that the lines of force of the gravitational field run in the negative direction of the axis of z. In a space free of gravitational fields let there be a second system of co-ordinates K, moving with uniform acceleration (g) in the positive direction of the z axis Relatively to K, as well as relatively to K, material points which are not subjected to the action of other material points, move in keeping with the equations

d2x/dt2 = 0, d2y/dt2 = 0, d2z/dt2 = -g

we arrive at a very satisfactory interpretation of this law of experience; if we assume that the system K and K are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being is a space free from gravitational fields, if we then regard K as uniformly accelerated. This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field a matter of course.

: The velocity of Light is a constant, light follows the shortest distance between two points. If light travels through a gravitational field, which it routinely does, it must accelerate. But the only way light can accelerate while appearing to maintain a constant speed is if the shortest distance between two points is NOT a straight line but a curve.

Actually that is true only for inertial frames of reference in flat spacetime (i.e. Minkowski frames). In gravitational fields the speed of light depends on the position of the light in the gravitational field. However when measured locally it's always "c" (as measured in an inertial frame). Near the Sun the speed of light is given by c' = (1 - 2GM/r) (G=c=1)

: So Mass causes Gravity which is an acceleration caused by the curvature of spacetime.

Acceleration is different than spacetime curvature. If you are in flat spacetime and then start to accelerate like if you were in a rocket chip, the spacetime in the new frame of reference would still be flat. The path of the particle is curved. Actually Einstein called acceleration "curvature" in the early days. The term has taken on different meaning in modern literature. It means what Newtonian physics calls a tidal gradient. Also the spacetime in a uniform gravitational field is flat. When a modern relativists speaks of gravity he is most likely refereing to curved spacetime. However this was never how Einstein thought of it. As a matter of fact Einstien didn't even like that interpretation. This is a common misunderstanding which has caused some problems. What Einstein said on this matter was

... what characterizes the existence of a gravitational field from the empirical standpoint is the non-vanishing of the [components of the affine connection], not the vanishing of the [components of the Riemann tensor]. If one does not think in such intuitive (anschaulich) ways, one cannot grasp why something like curvature should have anything at all to do with gravitation. In any case, no rational person would have hit upon anything otherwise. The key to the understanding of the equality of gravitational mass and inertial mass would have been missing.

: The curvature is caused by a mass.

More genererally it's caused by stress(e.g. pressure), energy and momentum. The mathematical quantity which describes this is called the stress-energy-momentum tensor. You may often see this called the energy-momentum tensor or the stress-energy tensor. But the proper name is the stress-energy-momentum tensor.

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